It is convenient for the purposes of the proof to replace the uniform random variables \(U_i\)  with \(1/U_i\) . Let \(Y_1, \dots , Y_N\)  be iid random variables with cdf \(F_Y(y)=1-1/y, y>1\) , \(F_Y(y)=0\)  for \(y\le 1\) . Observe that \(Y_i \overset{d}{=} 1/U_i\)  where \(U_i\)  is Uniform[0, 1].
Let \(Y_{1, N}\le \dots \le Y_{N, N}\)  be the order statistics, then \(U_{k, N} \overset{d}{=} 1/Y_{N-k, N}\) . As pointed out by [1]} (p. 50)
\(\theta _{N-k, N} & \overset{d}{=} U_F(Y_{N-k, N}), \\\vartheta _{N-k, N, T} & \overset{d}{=} U_T(Y_{N-k, N}).\) 
